20 Kinematics and dynamics of an expanding universe
a homogeneous, isotropic universe. In relativistic theory, there is no absolute time
and spatial distances are not invariant with respect to coordinate transformations.
Instead, the infinitesimal spacetime interval between events is invariant. There exist,
however, preferred coordinate systems in which the symmetries of the universe are
clearly manifest. In one of the most convenient of such coordinate systems, the
interval takes the form
ds^2 =dt^2 −dl 32 d=dt^2 −a^2 (t)(
dr^2
1 −kr^2+r^2 d
2)
≡gαβdxαdxβ, (1.47)wheregαβis the metric of the spacetime andxα≡(t,r,θ,φ)are the coordinates of
events. We will use the Einstein convention for summation over repeated indices:
gαβdxαdxβ≡∑
α,βgαβdxαdxβ.Additionally, we will always choose Greek indices to run from 0 to 3 with 0 re-
served for the time-like coordinate. Latin indices run only over spatial coordinates:
i,l,...= 1 , 2 ,3. The spatial coordinates introduced above are comoving; that is,
every object with zero peculiar velocity has constant coordinatesr,θ,φ.Further-
more, the coordinatetis the proper time measured by a comoving observer. The
distance between two comoving observers at a particular moment of time is
∫ √
−dst^2 =const∝a(t)
and, therefore, increases or decreases in proportion to the scale factor.
In General Relativity, the dynamical variables characterizing the gravitational
field are the components of the metricgαβ(xγ) and they obey the Einstein equations:
Gαβ≡Rαβ−1
2
δαβR−δβα= 8 πGTβα. (1.48)Here
Rβα=gαγ(
∂δγβ
∂xδ−
∂δγδ
∂xβ+γβδ δσσ −σγδβσδ)
(1.49)
is the Ricci tensor expressed in terms of the inverse metricgαγ, defined via
gαγgγβ=δαβ, and the Christoffel symbols
γβα =1
2
gαδ(
∂gγδ
∂xβ+
∂gδβ
∂xγ−
∂gγβ
∂xδ)
. (1.50)
The symbolδαβdenotes the unit tensor, equal to 1 whenα=βand 0 otherwise;
R=Rααis the scalar curvature; and= const is the cosmological term. Matter